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analysis: Erdős problems mapped to 4-primitive framework

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Identified 12 Erdős problems amenable to 4-primitive framework approach.

Primitive distribution:
- Packet: 6 problems (50%) - encoding/witness problems dominate
- Field: 2 problems (16.7%) - density/distribution problems
- Shear: 2 problems (16.7%) - extremal/metric problems
- Spectral: 2 problems (16.7%) - eigenvalue problems

High priority problems:
- Erdős–Rényi Random Graph Model (SPECTRAL) - eigenvalue distribution, VERY HIGH feasibility
- Erdős–Turán Conjecture (FIELD) - additive basis density, HIGH feasibility
- Erdős–Straus Conjecture (PACKET) - Egyptian fraction encoding, HIGH feasibility
- Erdős Conjecture on Arithmetic Progressions (FIELD) - density implies structure, HIGH feasibility

Recommended approach order:
1. Erdős–Rényi (validation point, spectral methods standard)
2. Erdős–Turán (additive basis density)
3. Erdős–Straus (Diophantine encoding)
4. Erdős Conjecture on APs (density implies structure)

Key insight: Packet primitive dominates - many Erdős problems are about
encodings/witness structures. All primitives represented - framework
covers diverse Erdős problem types.

Mapping saved to: 4-Infrastructure/shim/erdos_problems_4primitive_mapping.json
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{
"primitives": {
"field": {
"equation": "\u03c1(x\u20d7)",
"role": "tells you what exists (field / substrate / scalar manifold state)",
"erdos_applications": [
"Density of primes and prime gaps",
"Arithmetic progressions in dense sets (Szemer\u00e9di)",
"Distribution of integers in additive sets",
"Density in combinatorial structures",
"Erd\u0151s\u2013Tur\u00e1n theorem on additive bases"
]
},
"shear": {
"equation": "G = A\u1d40A",
"role": "tells you how it deforms (shear / metric deformation / lawful geometry)",
"erdos_applications": [
"Graph distances and metric embeddings",
"Extremal graph theory (max/min edges)",
"Graph isoperimetry and expansion",
"Erd\u0151s\u2013Stone theorem (extremal function)",
"Graph minor theory and treewidth"
]
},
"packet": {
"equation": "\u0393\u1d62",
"role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)",
"erdos_applications": [
"Ramsey numbers and witness structures",
"Extremal set systems (covering/packing)",
"Erd\u0151s\u2013Ko\u2013Rado theorem",
"Erd\u0151s\u2013Szekeres theorem (monotone subsequences)",
"Erd\u0151s\u2013Ginzburg\u2013Ziv theorem (zero-sum subsets)"
]
},
"spectral": {
"equation": "C = U\u039bU\u1d40",
"role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)",
"erdos_applications": [
"Graph spectra and eigenvalue bounds",
"Graph partitioning and clustering",
"Random graph eigenvalue distributions",
"Erd\u0151s\u2013R\u00e9nyi model properties",
"Expander graphs and spectral gap"
]
}
},
"erdos_problems": {
"high_priority": {
"erdos_turan_conjecture": {
"name": "Erd\u0151s\u2013Tur\u00e1n Conjecture on Additive Bases",
"statement": "If A is an additive basis of order 2 for the natural numbers, then the sum of reciprocals diverges: \u03a3_{a\u2208A} 1/a = \u221e",
"primitive": "field",
"mapping": "Additive basis density = field distribution. Conjecture about density of basis elements.",
"approach": "Treat A as density field \u03c1(n). Analyze spectral decomposition of additive structure. Use field primitive to model basis density and shear primitive to analyze additive deformation.",
"feasibility": "HIGH - Directly about density/distribution, maps cleanly to field primitive"
},
"erdos_straus_conjecture": {
"name": "Erd\u0151s\u2013Straus Conjecture",
"statement": "For every integer n \u2265 2, the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers x, y, z",
"primitive": "packet",
"mapping": "Egyptian fraction decomposition = packet encoding. Each solution is a packet (x,y,z) encoding 4/n.",
"approach": "Treat solutions as packets. Use packet primitive to search for encoding space. Spectral analysis of solution space structure.",
"feasibility": "HIGH - Problem about finding encodings/packets, natural fit for packet primitive"
},
"erdos_conjecture_arithmetic_progressions": {
"name": "Erd\u0151s Conjecture on Arithmetic Progressions",
"statement": "If \u03a3_{a\u2208A} 1/a diverges, then A contains arbitrarily long arithmetic progressions",
"primitive": "field",
"mapping": "Divergent reciprocal sum = high density field. High density implies rich structure (APs).",
"approach": "Use field primitive to model density \u03c1(A). Apply shear primitive to analyze how density deforms under translation (arithmetic progression structure). Spectral decomposition to detect periodic structure.",
"feasibility": "HIGH - Directly about density implying structure, field primitive natural fit"
},
"erdos_renyi_random_graph": {
"name": "Erd\u0151s\u2013R\u00e9nyi Random Graph Model",
"statement": "Study properties of G(n,p) random graphs. Threshold phenomena for connectivity, giant component, Hamiltonicity",
"primitive": "spectral",
"mapping": "Random graph eigenvalue distribution = spectral basis. Phase transitions = spectral pruning.",
"approach": "Use spectral primitive to analyze eigenvalue distribution of G(n,p). Detect phase transitions via spectral gap. Field primitive for density of edges.",
"feasibility": "VERY HIGH - Well-studied, spectral methods standard, direct mapping"
}
},
"medium_priority": {
"erdos_ko_rado": {
"name": "Erd\u0151s\u2013Ko\u2013Rado Theorem (extensions)",
"statement": "Maximum size of intersecting families of k-subsets of {1,...,n}",
"primitive": "packet",
"mapping": "Intersecting family = packet collection with witness property (intersection).",
"approach": "Treat each family as packet set. Use packet primitive to analyze encoding constraints. Spectral analysis of intersection graph.",
"feasibility": "MEDIUM - Solved for large n, but extensions open. Packet primitive useful for generalizations"
},
"erdos_szekeres": {
"name": "Erd\u0151s\u2013Szekeres Theorem (generalizations)",
"statement": "Any sequence of n\u00b2+1 distinct real numbers contains a monotone subsequence of length n+1",
"primitive": "packet",
"mapping": "Monotone subsequence = packet witness. Ramsey-type problem about finding structure.",
"approach": "Use packet primitive to model subsequences as witnesses. Shear primitive for ordering deformation. Spectral analysis of permutation patterns.",
"feasibility": "MEDIUM - Solved, but generalizations and extensions open"
},
"erdos_ginzburg_ziv": {
"name": "Erd\u0151s\u2013Ginzburg\u2013Ziv Theorem (extensions)",
"statement": "Any 2n-1 integers contain n whose sum is divisible by n",
"primitive": "packet",
"mapping": "Zero-sum subset = packet with witness property (sum = 0 mod n).",
"approach": "Treat subsets as packets. Use packet primitive to search for zero-sum encoding. Spectral analysis of additive structure modulo n.",
"feasibility": "MEDIUM - Solved, but extensions to other groups and structures open"
},
"erdos_stone": {
"name": "Erd\u0151s\u2013Stone Theorem (extremal function)",
"statement": "For any graph H, ex(n,H) = (1 - 1/\u03c7(H)-1 + o(1))n\u00b2/2 where \u03c7(H) is chromatic number",
"primitive": "shear",
"mapping": "Extremal function = shear metric. Maximum edges without H = deformation constraint.",
"approach": "Use shear primitive to analyze edge density under forbidden subgraph constraint. Spectral analysis of extremal graphs. Field primitive for density.",
"feasibility": "MEDIUM - Solved, but generalizations to hypergraphs open"
}
},
"exploratory": {
"erdos_faber_lovasz": {
"name": "Erd\u0151s\u2013Faber\u2013Lov\u00e1sz Conjecture",
"statement": "If each edge of a complete graph on n vertices is colored with one of n colors, then there exists a set of n edges with no two sharing a vertex or having the same color",
"primitive": "packet",
"mapping": "Edge coloring = packet encoding. Matching = packet set with witness properties.",
"approach": "Use packet primitive to model edge colorings as encodings. Spectral analysis of intersection graph. Shear for matching constraints.",
"feasibility": "EXPLORATORY - Recently solved (2021), but method could generalize"
},
"erdos_distinct_distances": {
"name": "Erd\u0151s Distinct Distances Problem",
"statement": "Any set of n points in the plane determines at least n/\u221alog n distinct distances",
"primitive": "shear",
"mapping": "Distance set = shear metric. Point configuration = field manifold.",
"approach": "Use field primitive for point configuration. Shear primitive for distance metric. Spectral analysis of distance distribution.",
"feasibility": "EXPLORATORY - Solved (Guth-Katz), but 4-primitive approach could provide new perspective"
},
"erdos_moser_problem": {
"name": "Erd\u0151s\u2013Moser Problem",
"statement": "Find all solutions to 1/a + 1/b + 1/c + 1/d + 1/e = 1 in distinct positive integers",
"primitive": "packet",
"mapping": "Egyptian fraction decomposition = packet encoding. Each solution is a 5-tuple packet.",
"approach": "Use packet primitive to search for encoding space. Spectral analysis of solution structure. Field for density of solutions.",
"feasibility": "EXPLORATORY - Solved (only known solution), but method could generalize to other Diophantine equations"
},
"erdos_hadamard": {
"name": "Erd\u0151s Hadamard Conjecture",
"statement": "There exist Hadamard matrices of order 4k for all k",
"primitive": "spectral",
"mapping": "Hadamard matrix = spectral basis (orthogonal rows/columns). Eigenvalues = \u00b1\u221an.",
"approach": "Use spectral primitive to analyze matrix structure. Field for existence density. Packet for construction methods.",
"feasibility": "EXPLORATORY - Open problem, spectral methods standard in Hadamard matrix theory"
}
}
},
"primitive_distribution": {
"field": 2,
"shear": 2,
"packet": 6,
"spectral": 2
},
"recommended_order": [
"erdos_renyi_random_graph",
"erdos_turan_conjecture",
"erdos_straus_conjecture",
"erdos_conjecture_arithmetic_progressions"
],
"insights": {
"packet_dominance": "Packet primitive dominates (5 problems) - many Erd\u0151s problems are about encodings/witnesses",
"spectral_validation": "Erd\u0151s\u2013R\u00e9nyi provides validation point - spectral methods standard",
"field_additive": "Field primitive for additive problems - density and structure",
"shear_extremal": "Shear primitive for extremal problems - metric and deformation",
"cross_domain": "All primitives represented - framework covers diverse Erd\u0151s problem types"
}
}

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#!/usr/bin/env python3
"""
Erdős Problems → 4-Primitive Framework Mapping
==============================================
Identify which Erdős problems could be approached using the
4-primitive framework (field, shear, packet, spectral).
"""
import json
from pathlib import Path
RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
# 4-primitive framework
PRIMITIVES = {
"field": {
"equation": "ρ(x⃗)",
"role": "tells you what exists (field / substrate / scalar manifold state)",
"erdos_applications": [
"Density of primes and prime gaps",
"Arithmetic progressions in dense sets (Szemerédi)",
"Distribution of integers in additive sets",
"Density in combinatorial structures",
"Erdős–Turán theorem on additive bases"
]
},
"shear": {
"equation": "G = AᵀA",
"role": "tells you how it deforms (shear / metric deformation / lawful geometry)",
"erdos_applications": [
"Graph distances and metric embeddings",
"Extremal graph theory (max/min edges)",
"Graph isoperimetry and expansion",
"Erdős–Stone theorem (extremal function)",
"Graph minor theory and treewidth"
]
},
"packet": {
"equation": "Γᵢ",
"role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)",
"erdos_applications": [
"Ramsey numbers and witness structures",
"Extremal set systems (covering/packing)",
"Erdős–Ko–Rado theorem",
"Erdős–Szekeres theorem (monotone subsequences)",
"Erdős–Ginzburg–Ziv theorem (zero-sum subsets)"
]
},
"spectral": {
"equation": "C = UΛUᵀ",
"role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)",
"erdos_applications": [
"Graph spectra and eigenvalue bounds",
"Graph partitioning and clustering",
"Random graph eigenvalue distributions",
"Erdős–Rényi model properties",
"Expander graphs and spectral gap"
]
}
}
# Erdős problems mapped to 4 primitives
ERDOS_PROBLEMS = {
"high_priority": {
"erdos_turan_conjecture": {
"name": "Erdős–Turán Conjecture on Additive Bases",
"statement": "If A is an additive basis of order 2 for the natural numbers, then the sum of reciprocals diverges: Σ_{a∈A} 1/a = ∞",
"primitive": "field",
"mapping": "Additive basis density = field distribution. Conjecture about density of basis elements.",
"approach": "Treat A as density field ρ(n). Analyze spectral decomposition of additive structure. Use field primitive to model basis density and shear primitive to analyze additive deformation.",
"feasibility": "HIGH - Directly about density/distribution, maps cleanly to field primitive"
},
"erdos_straus_conjecture": {
"name": "Erdős–Straus Conjecture",
"statement": "For every integer n ≥ 2, the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers x, y, z",
"primitive": "packet",
"mapping": "Egyptian fraction decomposition = packet encoding. Each solution is a packet (x,y,z) encoding 4/n.",
"approach": "Treat solutions as packets. Use packet primitive to search for encoding space. Spectral analysis of solution space structure.",
"feasibility": "HIGH - Problem about finding encodings/packets, natural fit for packet primitive"
},
"erdos_conjecture_arithmetic_progressions": {
"name": "Erdős Conjecture on Arithmetic Progressions",
"statement": "If Σ_{a∈A} 1/a diverges, then A contains arbitrarily long arithmetic progressions",
"primitive": "field",
"mapping": "Divergent reciprocal sum = high density field. High density implies rich structure (APs).",
"approach": "Use field primitive to model density ρ(A). Apply shear primitive to analyze how density deforms under translation (arithmetic progression structure). Spectral decomposition to detect periodic structure.",
"feasibility": "HIGH - Directly about density implying structure, field primitive natural fit"
},
"erdos_renyi_random_graph": {
"name": "Erdős–Rényi Random Graph Model",
"statement": "Study properties of G(n,p) random graphs. Threshold phenomena for connectivity, giant component, Hamiltonicity",
"primitive": "spectral",
"mapping": "Random graph eigenvalue distribution = spectral basis. Phase transitions = spectral pruning.",
"approach": "Use spectral primitive to analyze eigenvalue distribution of G(n,p). Detect phase transitions via spectral gap. Field primitive for density of edges.",
"feasibility": "VERY HIGH - Well-studied, spectral methods standard, direct mapping"
}
},
"medium_priority": {
"erdos_ko_rado": {
"name": "Erdős–Ko–Rado Theorem (extensions)",
"statement": "Maximum size of intersecting families of k-subsets of {1,...,n}",
"primitive": "packet",
"mapping": "Intersecting family = packet collection with witness property (intersection).",
"approach": "Treat each family as packet set. Use packet primitive to analyze encoding constraints. Spectral analysis of intersection graph.",
"feasibility": "MEDIUM - Solved for large n, but extensions open. Packet primitive useful for generalizations"
},
"erdos_szekeres": {
"name": "Erdős–Szekeres Theorem (generalizations)",
"statement": "Any sequence of n²+1 distinct real numbers contains a monotone subsequence of length n+1",
"primitive": "packet",
"mapping": "Monotone subsequence = packet witness. Ramsey-type problem about finding structure.",
"approach": "Use packet primitive to model subsequences as witnesses. Shear primitive for ordering deformation. Spectral analysis of permutation patterns.",
"feasibility": "MEDIUM - Solved, but generalizations and extensions open"
},
"erdos_ginzburg_ziv": {
"name": "Erdős–Ginzburg–Ziv Theorem (extensions)",
"statement": "Any 2n-1 integers contain n whose sum is divisible by n",
"primitive": "packet",
"mapping": "Zero-sum subset = packet with witness property (sum = 0 mod n).",
"approach": "Treat subsets as packets. Use packet primitive to search for zero-sum encoding. Spectral analysis of additive structure modulo n.",
"feasibility": "MEDIUM - Solved, but extensions to other groups and structures open"
},
"erdos_stone": {
"name": "Erdős–Stone Theorem (extremal function)",
"statement": "For any graph H, ex(n,H) = (1 - 1/χ(H)-1 + o(1))n²/2 where χ(H) is chromatic number",
"primitive": "shear",
"mapping": "Extremal function = shear metric. Maximum edges without H = deformation constraint.",
"approach": "Use shear primitive to analyze edge density under forbidden subgraph constraint. Spectral analysis of extremal graphs. Field primitive for density.",
"feasibility": "MEDIUM - Solved, but generalizations to hypergraphs open"
}
},
"exploratory": {
"erdos_faber_lovasz": {
"name": "Erdős–Faber–Lovász Conjecture",
"statement": "If each edge of a complete graph on n vertices is colored with one of n colors, then there exists a set of n edges with no two sharing a vertex or having the same color",
"primitive": "packet",
"mapping": "Edge coloring = packet encoding. Matching = packet set with witness properties.",
"approach": "Use packet primitive to model edge colorings as encodings. Spectral analysis of intersection graph. Shear for matching constraints.",
"feasibility": "EXPLORATORY - Recently solved (2021), but method could generalize"
},
"erdos_distinct_distances": {
"name": "Erdős Distinct Distances Problem",
"statement": "Any set of n points in the plane determines at least n/√log n distinct distances",
"primitive": "shear",
"mapping": "Distance set = shear metric. Point configuration = field manifold.",
"approach": "Use field primitive for point configuration. Shear primitive for distance metric. Spectral analysis of distance distribution.",
"feasibility": "EXPLORATORY - Solved (Guth-Katz), but 4-primitive approach could provide new perspective"
},
"erdos_moser_problem": {
"name": "Erdős–Moser Problem",
"statement": "Find all solutions to 1/a + 1/b + 1/c + 1/d + 1/e = 1 in distinct positive integers",
"primitive": "packet",
"mapping": "Egyptian fraction decomposition = packet encoding. Each solution is a 5-tuple packet.",
"approach": "Use packet primitive to search for encoding space. Spectral analysis of solution structure. Field for density of solutions.",
"feasibility": "EXPLORATORY - Solved (only known solution), but method could generalize to other Diophantine equations"
},
"erdos_hadamard": {
"name": "Erdős Hadamard Conjecture",
"statement": "There exist Hadamard matrices of order 4k for all k",
"primitive": "spectral",
"mapping": "Hadamard matrix = spectral basis (orthogonal rows/columns). Eigenvalues = ±√n.",
"approach": "Use spectral primitive to analyze matrix structure. Field for existence density. Packet for construction methods.",
"feasibility": "EXPLORATORY - Open problem, spectral methods standard in Hadamard matrix theory"
}
}
}
def analyze_erdos_mapping():
print("=" * 70)
print(" ERDŐS PROBLEMS → 4-PRIMITIVE FRAMEWORK MAPPING")
print("=" * 70)
print("\n4-PRIMITIVE FRAMEWORK:")
for prim, data in PRIMITIVES.items():
print(f"\n{prim.upper()}: {data['equation']}")
print(f" Role: {data['role']}")
print(f" Erdős applications:")
for app in data['erdos_applications']:
print(f" • {app}")
print("\n" + "=" * 70)
print(" ERDŐS PROBLEMS BY PRIORITY")
print("=" * 70)
for priority, problems in ERDOS_PROBLEMS.items():
print(f"\n{priority.upper().replace('_', ' ')} ({len(problems)} problems):")
for prob_id, prob in problems.items():
prim = prob["primitive"].upper()
print(f"\n • {prob['name']}")
print(f" Primitive: {prim}")
print(f" Statement: {prob['statement'][:100]}...")
print(f" Mapping: {prob['mapping']}")
print(f" Feasibility: {prob['feasibility']}")
print("\n" + "=" * 70)
print(" PRIMITIVE DISTRIBUTION")
print("=" * 70)
primitive_counts = {"field": 0, "shear": 0, "packet": 0, "spectral": 0}
for priority, problems in ERDOS_PROBLEMS.items():
for prob in problems.values():
primitive_counts[prob["primitive"]] += 1
total = sum(primitive_counts.values())
for prim, count in primitive_counts.items():
percent = count / total * 100 if total > 0 else 0
print(f"\n{prim.upper()} ({count} problems, {percent:.1f}%):")
problems_list = []
for priority, problems in ERDOS_PROBLEMS.items():
for prob_id, prob in problems.items():
if prob["primitive"] == prim:
problems_list.append(f"{prob['name']} ({priority})")
for p in problems_list:
print(f" • {p}")
print("\n" + "=" * 70)
print(" RECOMMENDED APPROACH ORDER")
print("=" * 70)
print("\n1. Erdős–Rényi Random Graph Model (SPECTRAL)")
print(" - Why: Well-studied, spectral methods standard, direct mapping")
print(" - Approach: Analyze eigenvalue distribution of G(n,p), detect phase transitions via spectral gap")
print(" - Expected outcome: New insights into random graph phase transitions")
print("\n2. Erdős–Turán Conjecture (FIELD)")
print(" - Why: Directly about density/distribution, maps cleanly to field primitive")
print(" - Approach: Treat additive basis as density field, analyze spectral decomposition of additive structure")
print(" - Expected outcome: New perspective on basis density and additive structure")
print("\n3. Erdős–Straus Conjecture (PACKET)")
print(" - Why: Problem about finding encodings/packets, natural fit for packet primitive")
print(" - Approach: Treat solutions as packets, search encoding space, spectral analysis of solution structure")
print(" - Expected outcome: Potential progress on long-standing Diophantine problem")
print("\n4. Erdős Conjecture on Arithmetic Progressions (FIELD)")
print(" - Why: Directly about density implying structure, field primitive natural fit")
print(" - Approach: Model density ρ(A), apply shear to analyze density deformation under translation")
print(" - Expected outcome: New approach to Szemerédi-type theorems")
print("\n" + "=" * 70)
print(" KEY INSIGHTS")
print("=" * 70)
print("\n1. Field primitive (2 problems):")
print(" - Erdős–Turán Conjecture: additive basis density")
print(" - Erdős Conjecture on APs: density implies structure")
print(" - Core: density problems, distribution analysis, additive structure")
print("\n2. Shear primitive (2 problems):")
print(" - Erdős–Stone Theorem: extremal function")
print(" - Erdős Distinct Distances: distance metric")
print(" - Core: extremal problems, metric geometry, graph deformation")
print("\n3. Packet primitive (5 problems):")
print(" - Erdős–Straus Conjecture: Egyptian fraction encoding")
print(" - Erdős–Ko–Rado: intersecting families")
print(" - Erdős–Szekeres: monotone subsequences")
print(" - Erdős–Ginzburg–Ziv: zero-sum subsets")
print(" - Erdős–Faber–Lovász: edge colorings")
print(" - Core: encoding problems, witness structures, Ramsey-type problems")
print("\n4. Spectral primitive (2 problems):")
print(" - Erdős–Rényi Random Graph: eigenvalue distribution")
print(" - Erdős Hadamard Conjecture: orthogonal matrices")
print(" - Core: eigenvalue problems, spectral methods, random matrix theory")
print("\n5. Cross-domain patterns:")
print(" - Packet primitive dominates (5 problems) - many Erdős problems are about encodings/witnesses")
print(" - Field and spectral each have 2 problems - density and eigenvalue problems are common")
print(" - Shear has 2 problems - extremal and metric geometry problems")
print(" - All primitives represented - 4-primitive framework covers diverse Erdős problem types")
print("\n6. Recommended starting point:")
print(" - Erdős–Rényi Random Graph Model: spectral methods standard, high feasibility")
print(" - This provides a validation of the 4-primitive framework on well-understood problem")
print(" - Success here would validate approach for harder problems (Erdős–Turán, Erdős–Straus)")
# Save mapping
output_file = RESEARCH_STACK / "4-Infrastructure/shim/erdos_problems_4primitive_mapping.json"
with open(output_file, 'w') as f:
json.dump({
"primitives": PRIMITIVES,
"erdos_problems": ERDOS_PROBLEMS,
"primitive_distribution": primitive_counts,
"recommended_order": [
"erdos_renyi_random_graph",
"erdos_turan_conjecture",
"erdos_straus_conjecture",
"erdos_conjecture_arithmetic_progressions"
],
"insights": {
"packet_dominance": "Packet primitive dominates (5 problems) - many Erdős problems are about encodings/witnesses",
"spectral_validation": "Erdős–Rényi provides validation point - spectral methods standard",
"field_additive": "Field primitive for additive problems - density and structure",
"shear_extremal": "Shear primitive for extremal problems - metric and deformation",
"cross_domain": "All primitives represented - framework covers diverse Erdős problem types"
}
}, f, indent=2)
print(f"\n✓ Mapping saved to: {output_file}")
if __name__ == "__main__":
analyze_erdos_mapping()